Dispersive processing of borehole acoustic data is useful for the characterization and estimation of rock properties using borehole acoustic data containing propagating modes. It is both more complex and more informative than non-dispersive processing, where bulk velocities in the rock are measured without any frequency dependence. Perhaps the most common parameters used to describe the dispersion characteristics are the wavenumber, k(f), and the attenuation, A(f), both of which are functions of the frequency f and are of interest in the oil industry for characterizing the properties of reservoirs. The dispersion characteristics include the group and phase slowness (reciprocal of velocity) as a function of frequency linked to the wavenumber, k(f), as follows:
                                                        s              ϕ                        ⁡                          (              f              )                                =                                    1                                                V                  phase                                ⁡                                  (                  f                  )                                                      =                                          k                ⁡                                  (                  f                  )                                            f                                      ⁢                                  ⁢        and                            (        1        )                                                      s            g                    ⁡                      (            f            )                          =                              1                                          V                group                            ⁡                              (                f                )                                              =                                    ⅆ                              k                ⁡                                  (                  f                  )                                                                    ⅆ              f                                                          (        2        )            These two quantities are not independent, and satisfy:
                              s          g                =                              s            ϕ                    +                      f            ⁢                                                            ⅆ                                      s                    ϕ                                                                    ⅆ                  f                                            .                                                          (        3        )            
One known class of dispersive processing solutions uses physical models which relate the rock properties around the borehole to predicted dispersion curves. Waveform data collected by an array of sensors is backpropagated according to the modeled dispersion curves, and the model is adjusted until there is good semblance among these backpropagated waveforms, thereby indicating a good fit of the model to the data. However, models are presently available for only simpler cases, and other physical parameters still need to be known. Further, there is a risk that biased results will be produced if there is model mismatch or input parameter error. Moreover, this class of solutions assumes the presence of only a single modeled propagating mode, and pre-processing steps such as time windowing and filtering may be desireable to isolate the mode of interest. These also require user input, and in some cases the results may be sensitive to the latter, requiring expert users for correctly processing the data.
One way to mitigate these drawbacks is to directly estimate the dispersion characteristics from logging data, i.e., without reference to particular physical models. Not only can this be used for quantitative inversion of parameters of interest, but the dispersion curves carry important information about the acoustic state of the rock and are important tools for interpretation and validation. Moreover, in order to be part of a commercial processing chain, the dispersion estimation method should be capable of operation in an automated unsupervised manner and in particular the accuracy of the results should not depend on input from highly skilled users. One technique of estimating dispersion characteristics directly from data collected by an array of sensors, for example, in seismic applications, is to use a 2D FFT, also called the f-k (frequency wavenumber) transform. This technique indicates the dispersion characteristics of propagating waves, both dispersive and non-dispersive. However, its effectiveness is limited to large arrays of tens of sensors. For applications with fewer sensors, such as commonly found in sensor arrays (2-13 sensors) on borehole wireline and LWD tools, this technique may lack the necessary resolution and accuracy to produce useful answers.
A high resolution method appropriate for shorter arrays was developed using narrow band array processing techniques applied to frequency domain data obtained by performing an FFT on the array waveform data. However, while this is an effective tool for studying dispersion behavior in a supervised setting with user input, it produces unlabelled dots on the f-k plane frequency by frequency and therefore may not be entirely suitable for deployment as an automated unsupervised processing method. Moreover, the operation of the FFT washes out the information pertaining to the time of arrival of various propagating modes, thereby compromising performance and the effectiveness of interpretation, especially for weaker modes of interest overlapping with stronger ones in frequency domain.
A parametric method for estimating dipole flexural dispersion is known which can be used for automated dispersion extraction. However this is limited to the particular (flexural) dispersive mode, and cannot be readily extended to the general case. Moreover, it is a one-component extraction technique.
Recently, a new algorithm has been proposed using time frequency analysis with continuous wavelet transform along with beamforming methods. However, the algorithm does not account for attenuation and has a propagation model with a pure phase response. In real world applications, many modes of interest are attenuative and the attenuation is therefore a key parameter of interest. Further, the phase response is estimated using a 2-D search over delays and phase corrections applied to the original data, for each of which, a time-frequency map is generated. Extending this approach to include attenuation would make it a 3-D search, which would be relatively computationally intensive for commercial real time applications. Finally, the use of this approach as well as the criterion of maximum energy for identifying propagating modes on the time-frequency maps generated above may obscure weak modes of interest by stronger in-band interfering modes even when they are separated in time.